Conjecture on the limiting distribution of normalized interface-curve length

Let $\Gamma$ be the interface curve joining opposite corners of a three-dimensional cube of size $N$, and let $L_N$ be its length, defined as the number of tetrahedra crossed by the curve. Write $\widehat{L}_N$ for the length normalized by its average. Uniform-limit conjecture. The random variable $\widehat{L}_N$ converges in law to the uniform distribution on $[0,2]$. The observed average growth $\mathbb{E}[L_N]\sim N^3$ suggests that the interface curve is space-filling; the proposed limiting law is supported by numerical data, but finite-size effects remain near $0$ and $2$.